796 research outputs found
Independence-friendly cylindric set algebras
Independence-friendly logic is a conservative extension of first-order logic
that has the same expressive power as existential second-order logic. In her
Ph.D. thesis, Dechesne introduces a variant of independence-friendly logic
called IFG logic. We attempt to algebraize IFG logic in the same way that
Boolean algebra is the algebra of propositional logic and cylindric algebra is
the algebra of first-order logic.
We define independence-friendly cylindric set algebras and prove two main
results. First, every independence-friendly cylindric set algebra over a
structure has an underlying Kleene algebra. Moreover, the class of such
underlying Kleene algebras generates the variety of all Kleene algebras. Hence
the equational theory of the class of Kleene algebras that underly an
independence-friendly cylindric set algebra is finitely axiomatizable. Second,
every one-dimensional independence-friendly cylindric set algebra over a
structure has an underlying monadic Kleene algebra. However, the class of such
underlying monadic Kleene algebras does not generate the variety of all monadic
Kleene algebras. Finally, we offer a conjecture about which subvariety of
monadic Kleene algebras the class of such monadic Kleene algebras does
generate.Comment: 42 pages. Submitted to the Logic Journal of the IGPL. See also
http://math.colgate.edu/~amann
Lattice initial segments of the hyperdegrees
We affirm a conjecture of Sacks [1972] by showing that every countable
distributive lattice is isomorphic to an initial segment of the hyperdegrees,
. In fact, we prove that every sublattice of any
hyperarithmetic lattice (and so, in particular, every countable locally finite
lattice) is isomorphic to an initial segment of . Corollaries
include the decidability of the two quantifier theory of
and the undecidability of its three quantifier theory. The key tool in the
proof is a new lattice representation theorem that provides a notion of forcing
for which we can prove a version of the fusion lemma in the hyperarithmetic
setting and so the preservation of . Somewhat surprisingly,
the set theoretic analog of this forcing does not preserve . On
the other hand, we construct countable lattices that are not isomorphic to an
initial segment of
On the Failure of Fixed-Point Theorems for Chain-complete Lattices in the Effective Topos
In the effective topos there exists a chain-complete distributive lattice
with a monotone and progressive endomap which does not have a fixed point.
Consequently, the Bourbaki-Witt theorem and Tarski's fixed-point theorem for
chain-complete lattices do not have constructive (topos-valid) proofs
Scopes and Limits of Modality in Quantum Mechanics
We develop an algebraic frame for the simultaneous treatment of actual and
possible properties of quantum systems. We show that, in spite of the fact that
the language is enriched with the addition of a modal operator to the
orthomodular structure, contextuality remains a central feature of quantum
systems.Comment: 9 pages, no figure
Free -distributive lattice over an -element chain
In this note we provide an explicit construction of , the free -distributive lattice over an -element chain, different from those given by Cignoli [4] and Abad--Díaz Varela [1], and prove that can be endowed with a structure of a De Morgan algebra.Fil: Monteiro, Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Abad, Manuel. Universidad Nacional del Sur. Departamento de Matemática; ArgentinaFil: Zander, Marta Amalia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Matemática; Argentin
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